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Is the Abel-Ruffini Theorem essentially equivalent to saying that the set of all complex numbers constructable by a concatenation of field operations and n-roots of rational numbers is a proper subset of the algebraic numbers?

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I wouldn't put it quite that way. After all the Abel-Ruffini Theorem also tells us that in general the zeros of a fifth degree (or higher) polynomial are not contained in a root tower extension field of the smallest field containing the coefficients of the polynomials.

But that field generated by the coefficients of the given polynomial may very well contain transcendental numbers as well. If $K$ is the field generated by the coefficients of such a polynomial Abel-Ruffini says that the zeros of that polynomial, while necessarily transcendental numbers themselves, do not even belong to a root tower extension of $K$.

Of course, you are correct in stating that Abel-Ruffini also implies the existence of algebraic integers not contained in a root tower extension of the rationals. My point is that the scope of Abel-Ruffini is wider, and not restricted to subsets of the field of algebraic numbers.

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  • $\begingroup$ Oh, so for algebraic numbers we think of polynomials in $\mathbb{Q}[x]$, but for Abel Ruffini the polynomials can have coefficients in any field? $\endgroup$ – Robearz Jun 24 '14 at 18:41
  • $\begingroup$ Correct. I guess for the purposes of quintic polynomials we may want the field to have characteristic $>5$, but the field may, for example, be an extension of $\Bbb{C}$. $\endgroup$ – Jyrki Lahtonen Jun 24 '14 at 18:49

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